INVARIANTS OF MEASURE AND CATEGORY

@article{Bartoszynski2010INVARIANTSOM,
  title={INVARIANTS OF MEASURE AND CATEGORY},
  author={Tomek Bartoszynski},
  journal={arXiv: Logic},
  year={2010},
  pages={491-555}
}
The purpose of this chapter is to discuss various results concerning the relationship between measure and category. The focus is on set-theoretic properties of the associated ideals, particularly, their cardinal characteristics. The key notion is the Tukey reducibility which compares partial orders with respect to their cofinality type. We define small sets of reals associated with cardinal invariants and discuss their properties. We present a number of ZFC results and forcing-like… 
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References

SHOWING 1-10 OF 76 REFERENCES
Reductions Between Cardinal Characteristics of the Continuum
We discuss two general aspects of the theory of cardinal characteristics of the continuum, especially of proofs of inequalities between such characteristics. The first aspect is to express the
On two-cardinal properties of ideals
We investigate two-cardinal properties of ideals. These properties involve notions such as Luzin sets, special coverings, etc. We apply our results to the ideals of meagre sets and of negligible sets
Some properties of measure and category
Several elementary cardinal properties of measure and category on the real line are studied. For example, one property is that every set of real numbers of cardinality less than the continuum has
Analytic Ideals and Cofinal Types
The Structure of σ-Ideals of Compact Sets
Motivated by problems in certain areas of analysis, like measure theory and harmonic analysis, where σ-ideals of compact sets are encountered very often as notions of small or exceptional sets, we
THE CATEGORY OF COFINAL TYPES. I
Introduction. Briefly, this paper introduces a category which we call the category of cofinal types. We construct a concrete representation il of X', and we determine explicitly the part of i1
Generic constructions of small sets of reals
Norms on Possibilities I: Forcing With Trees and Creatures
We present a systematic study of the method of "norms on possibilities" of building forcing notions with keeping their properties under full control. This technique allows us to answer several open
A model of set-theory in which every set of reals is Lebesgue measurable*
We show that the existence of a non-Lebesgue measurable set cannot be proved in Zermelo-Frankel set theory (ZF) if use of the axiom of choice is disallowed. In fact, even adjoining an axiom DC to ZF,
Analytic Ideals and Their Applications
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